Derivative of log(x^5,3)

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   / 53\
   | --|
   | 10|
log\x  /

$$\log{\left(x^{\frac{53}{10}} \right)}$$

  /   / 53\\
  |   | --||
d |   | 10||
--\log\x  //
dx

$$\frac{d}{d x} \log{\left(x^{\frac{53}{10}} \right)}$$

Transform

Detail solution

Let $u = x^{\frac{53}{10}}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} x^{\frac{53}{10}}$ :
1. Apply the power rule: $x^{\frac{53}{10}}$ goes to $\frac{53 x^{\frac{43}{10}}}{10}$
The result of the chain rule is:

$\frac{53}{10 x}$

The answer is:

$\frac{53}{10 x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 53 
----
10*x

$$\frac{53}{10 x}$$

Simplify

The second derivative [src]

 -53 
-----
    2
10*x

$$- \frac{53}{10 x^{2}}$$

Simplify

The third derivative [src]

 53 
----
   3
5*x

$$\frac{53}{5 x^{3}}$$

Simplify

The graph